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Consider the following statements. "If x is an integer then 3+2=5" and "If x is not integer then 3+2=5". Constructing truth tables for the above statements show that there is no case P is true and Q is false. So both statements are true.

Also the statement where we have substitute the above statements "If (If P then Q and If not P then Q) then Q" is true. Can we say that the whole implication or Q is a tautology? I would say no because it depends on what we have defined as "integer", "3+2=5" etc. But from the above statements we can concluded that Q is true regardless of P i.e. always true. Does it make it a tautology? Can someone help me how to distinguish them?

From what I understand (might be wrong) tautologies are about truth tables irrespective of the meaning of the statements whereas theorems are based on the meaning of the statements. Another example is the statement "If x is positive then x squared is also positivie". It is true because we can eliminate from the truth table the line with (T and F) i.e. always true but not a tautology. But also when we "search" for tautologies we search for always true statements. Do they have in common the fact that both theorems (given the set of axioms in a system) and tautologies (given the set of axioms of laws of logic) are statements that are always true?

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    Tautologies are theorems of propositional logic with no extra axioms, but in general the answer is no. Even in predicate logic they are only a subset of all theorems (= logically valid formulas), namely those that can be obtained by substituting into propositional tautologies, see Tautology (logic). – Conifold Jul 21 at 4:14
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    Tautology applies to propositional calculus. In natural language, it has little sense to say that a statement S is a tautology (in the formal sense) because it is not very useful to apply the condition of the definition: "true in every possible interpretation." – Mauro ALLEGRANZA Jul 21 at 7:05
  • For example, a=a is Logically necessary in FOL, but it's not a Tautology, since it gives truth table TF. – Manx Jul 21 at 7:15
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    Am I missing something? The truth of "3+2=5" is independent of whether x is an integer or not. How are are either of the conclusions in "If x is an integer then 3+2=5" and "If x is not integer then 3+2=5" valid logically? – CJ Dennis Jul 21 at 10:40
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    @Manx What is "a", a variable? Then this is an open formula which has no truth value. And if it is a constant then what is supposed to be the "truth table" of a=a and how is it supposed to produce F? – Conifold Jul 21 at 14:30
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Considering the most simple case of mathematical theorems and logical tautologies:

  1. Theorems in mathematics are analytical true statements. They have the form “If A holds, then also B holds”.

    Example: “If two triangles have in common one side and the same two adjoining angles (A), then the triangles are congruent (B).”

    To prove a mathematical theorem means to discover from the definition of the concepts in A the property B. Therefore the proof is analytical.

  2. A tautology of propositional logic is a logical formula F(A,B,…) with variables A,B,…, such that: When replacing the variables by arbitrary statements then the resulting proposition is true.

    Example: If A implies B then non-B implies non-A.

Hence both concepts, tautology and mathematical theorem, are not the same. But it is interesting to elaborate on their difference.

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  • This is true, but there is a way to state any theorem as a tautology, although not the other way around. – Don Thousand Jul 21 at 14:48
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    Could you please state my theorem above as a tautology, thank you. – Jo Wehler Jul 21 at 14:58
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    By "analytical", do you mean "analytic" in Kant's sense? If so, there's some dispute over whether the a priori knowledge mathematics gives us is analytic or synthetic, and defending either position in that debate requires discussing the modern distinction mathematics makes between explicit and implicit definitions. – J.G. Jul 21 at 17:05
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    I think what @DonThousand means is that "If A holds, then also B holds” can be stated as a tautology "A -> B" – agemO Jul 22 at 10:45
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    "A => B" is not a tautology but an implication. The implication is not true for all propositions A and B. – Jo Wehler Jul 22 at 12:54
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Tautology applies to propositional logic:

a formula that is always true regardless of which valuation is used for the propositional variables.

The corresponding terms for predicate logic is that of Valid formula:

a formula that is true under every possible interpretation.

According to the definitions, a tautology is a valid formula of propositional logic.

In natural language, it has little sense to say that a statement S is a tautology (in the formal sense) because it is not very useful to apply the condition of the definition: "true in every possible interpretation."

From a formal point of view, a tautology is a theorem of propositional calculus.

A valid first order formula is a theorem of predicate calculus.

For formal arithmetic, i.e. the first-order version of Peano's axioms, a formula like e.g. 2+3=5 is a theorem, because it is provable from the axioms.

The arithmetical formula 2+3=5 is not valid, because it is not true in every interpretation.

But it is a Logical consequence of the axioms of arithmetic, because it is true in every interpretation that satisfies the axioms.

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Logicism, viz. the idea that mathematical statements are logical statements, has collapsed about a 100 years ago. Its most resolute advocates, Gottlob Frege (1848 -- 1925) and Bertrand Russell (1872 -- 1970) failed at building a consistent (Frege) and formally precise (Russell) derivation of mathematical truths from logical truths.

The nowadays orthodox foundations of mathematics work with axioms which cannot be held logically true. Take for instance the "axiom of empty set" in the Zermelo-Fraenkel axiomatic (ZF): "There is an empty set". There is nothing logical about that.

"3+2=5" is true in axiomatic mathematical systems like the ZF or the Peano axiomatic. Yet, neither are these system's axioms logically true, nor is "3+2=5".

Be P any statement and be Q=(3+2=5). Then "If P, than Q" is true, whatever P. But, it is not logically true, because, from a logical point of view, Q could be false.

If not logically true, mathematical propositions often are/were considered analytically true: viz. true by virtue of the concepts involved. "3+2=5" would in this sense be analytic by virtue of what 3, 2, 5 and = are or rather what "3", "2", "5" and "=" mean.

However, the distinction between analytic and non-analytic (synthetic) truths has been under heavy criticism ever since W.V.O. Quine's seminal essay "Two dogmas of empiricism" (1951). If this criticism holds, "3+2=5" is not systematically different concerning truth from the physicist's law of the conservation of energy.

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    "from a logical point of view, Q could be false." Except Q is not false. We just said it's not. So that "logical point of view" is unhelpful and wrong. I'd hate to see a philosopher try to do a jigsaw puzzle. "Oh, this piece fits here. Except, maybe it doesn't, because what if it were a different piece?" – Carl Leth Jul 21 at 17:51
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Your statements are indeed both tautologies. Thanks to Mauro Allegranza for finding the most straightforward definition:

a formula that is always true regardless of which valuation is used for the propositional variables.

So we have a way to check if a statement is a tautology:

  1. Find all the propositional variables
  2. Evaluate the truth of the statement for every possible assignment of true/false among those variables.

If the statement is true for every combination of those variables, it is a tautology.

If x is an integer then 3+2=5

  1. There are zero propositional variables.
  2. There is one possible way to assign true/false to zero variables: the empty function.

Is the statement true after performing the requisite zero variable assignments? Yes: the statement is unchanged, and as you pointed out, it happens to be true. Therefore the statement is true for every possible assignment and is a tautology. The same holds for the second statement.

Let's consider the statement "If P then 3+2=5". Is this a tautology? Now we have non-trivial work to do in the assignment stage, because we have one propositional variable. We check P = true and P = false, and see that this (different) statement is also a tautology.

it depends on what we have defined as "integer", "3+2=5" etc

No, this is an implicit part of the statement. If we don't have fixed definitions of those terms, then we have not made a logical statement yet. The definitions of the words used is an intrinsic part of the statement. If I were to change the definition of the symbol "3", I would be making a different logical statement and we'd have to evaluate that separately.

For the sake of argument, we are all implicitly agreeing to use the standard, common definitions of those symbols. If you secretly change those definitions without telling the people you are debating/arguing/reasoning with, then you are not acting in good faith. Similarly, if you're using a non-standard definition of one of those words or symbols, you must define them when making the statement. Otherwise you risk ambiguity and a frustrating argument where the parties don't realize they're arguing about different statements.

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    Can the downvoter please explain what's wrong with this? – Carl Leth Jul 21 at 18:19
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    Is it really this site's assertion that the truth of every single statement is ambiguous, because it's always possible for someone to have a different interpretation of the statement? Any logical statement written in French cannot be true or false because the reader might not speak French? The logical nature of the statement surely must be independent of whether it's represented in ASCII or UTF-8. Why are people on this site so insistent on challenging the definition of "3"? Either we agree on what a statement means or we give up on ever reasoning about logic. – Carl Leth Jul 21 at 19:40
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    So now the downvote makes sense: an outsider dared to trod on your turf and answer a simple question with a simple, logical derivation of a simple definition! The nerve! – Carl Leth Jul 21 at 21:46
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    @ClydeFrog how about instead of an ad hominem, territorial attack on me, you point out a single thing you disagree with in my answer? – Carl Leth Jul 21 at 21:52
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    A tautology in the strict sense is a logical truth of the propositional calculus. "2+3=5" is not a logical truth of the propositional calculus. Indeed, unless you subscribe to some form of logicism, it is not a logical truth at all, but a theorem of a theory of arithmetic, such as PA. To say that "2+3=5" might be false under some non-standard interpretation is not changing definitions, it is simply making use of an important distinction in logic between sentences that are true under some interpretation and sentences that are true under all interpretations. Only the latter are logical truths. – Bumble Jul 23 at 5:19

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